An inverse modeling procedure to determine particle growth and nucleation rates from measured aerosol size distributions

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An inverse modeling procedure to determine particle

growth and nucleation rates from measured aerosol size

distributions

B. Verheggen, M. Mozurkewich

To cite this version:

B. Verheggen, M. Mozurkewich. An inverse modeling procedure to determine particle growth and nucleation rates from measured aerosol size distributions. Atmospheric Chemistry and Physics Dis-cussions, European Geosciences Union, 2006, 6 (2), pp.1679-1723. �hal-00301055�

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Determination of particle growth and

nucleation rates B. Verheggen and M. Mozurkewich Title Page Abstract Introduction Conclusions References Tables Figures J I J I Back Close

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Printer-friendly Version Interactive Discussion Atmos. Chem. Phys. Discuss., 6, 1679–1723, 2006

www.atmos-chem-phys-discuss.net/6/1679/2006/ © Author(s) 2006. This work is licensed

under a Creative Commons License.

Atmospheric Chemistry and Physics Discussions

An inverse modeling procedure to

determine particle growth and nucleation

rates from measured aerosol size

distributions

B. Verheggen1,*and M. Mozurkewich1

1

Department of Chemistry and Centre for Atmospheric Chemistry, York University, Toronto ON M3J 1P3, Canada

*

now at: Laboratory for Atmospheric Chemistry, Paul Scherrer Institute, 5232 Villigen PSI, Switzerland

Received: 7 December 2005 – Accepted: 19 January 2006 – Published: 7 March 2006 Correspondence to: M. Mozurkewich (mozurkew@yorku.ca)

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Printer-friendly Version Interactive Discussion

Abstract

Classical nucleation theory is unable to explain the ubiquity of nucleation events ob-served in the atmosphere. This shows a need for an empirical determination of the nucleation rate. Here we present a novel inverse modeling procedure to determine par-ticle nucleation and growth rates based on consecutive measurements of the aerosol

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size distribution. The particle growth rate is determined by regression analysis of the measured change in the aerosol size distribution over time, taking into account the effects of processes such as coagulation, deposition and/or dilution. This allows the growth rate to be determined with a higher time-resolution than can be deduced from inspecting contour plots (“banana-plots”). Knowing the growth rate as a function of

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time enables the evaluation of the time of nucleation of measured particles of a certain size. The nucleation rate is then obtained by integrating the particle losses from time of measurement to time of nucleation. The regression analysis can also be used to deter-mine or verify the optimum value of other parameters of interest, such as the wall loss or coagulation rate constants. As an example, the method is applied to smog

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ber measurements. This program offers a powerful interpretive tool to study empirical aerosol population dynamics in general, and nucleation and growth in particular.

1 Introduction

From the many observations of new particle formation at different locations over the globe, it is now recognized that particle nucleation occurs widely in the troposphere

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(Kulmala et al., 2004). Small particles have adverse health effects (Oberdorster, 2001). When these particles grow to larger sizes they can directly affect climate by contributing to light scattering and absorption (Schwartz, 1996) and can indirectly affect climate by acting as cloud condensation nuclei and, therefore, altering cloud radiative properties and cloud lifetime (Lohmann and Feichter, 2005).

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Nucleation can occur in almost any environment, subject to a favourable set of con-1680

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Printer-friendly Version Interactive Discussion ditions. These conditions include a strong source of condensable vapour, high UV

ra-diation intensity, low aerosol surface area, high relative humidity, low temperature, and atmospheric mixing processes. It is noteworthy that of these locations, only nucleation in the free troposphere and in the vicinity of clouds seems to agree with predictions based on classical nucleation theory (Clarke et al., 1999).

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It has been suggested that the diameter of the critical cluster, the smallest parti-cle size for which the rate of condensation is larger than the rate of evaporation, is thought to be on the order of 1 nm (Weber et al., 1997; Kulmala et al., 2000). This cluster of a few molecules can hardly be described as being in the liquid phase, nor is it a gas. Even the use of the term “diameter” for this agglomerate of molecules

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is questionable (Preining, 1998). The classical nucleation theory, however, uses bulk liquid properties to describe the critical cluster and calculate the nucleation rate. Not surprisingly, there are large discrepancies between measurements (both laboratory and atmospheric) and classical nucleation theory; the discrepancy often amounts to several orders of magnitude (Wyslouzil et al., 1991; Weber et al., 1995, 1997, 1998;

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Andronache et al., 1997). Different parameterizations of nucleation rates give orders of magnitude different results (Kulmala and Laaksonen, 1990). Good agreement of theoretical nucleation rates with laboratory experiments has been presented (Viisanen et al., 1997), but the sulfuric acid concentrations used were much higher than is typi-cal for the atmosphere. Since the nucleation rate is extremely sensitive to the sulfuric

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acid concentration (Easter and Peters, 1994), extrapolation to atmospheric values is highly uncertain. These discrepancies illustrate a need to empirically determine the nucleation rate from measurements, independent of theory.

In a spatially and temporally homogeneous situation the average growth rate can be deduced from the time delay between the increase in precursor concentration and

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ultrafine particle number (Weber et al., 1997). If the location of the precursor source is known and constant in time, the inferred transport time to the measurement site can be used to infer the growth rate (Weber et al., 1998; O’Dowd et al., 1999), although dilution and coagulation would have influenced the size distribution during the time

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Printer-friendly Version Interactive Discussion between nucleation and measurement. These ways of determining the growth rate

are limited by the special conditions they require, and they provide an estimate of the growth rate averaged over relatively long time scales.

Usually the growth rate is estimated from the evolution of the maximum particle num-ber in the size distributions under homogeneous conditions by fitting the trajectory of

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highest particle concentration in a contour-plot of diameter versus time (M ¨akel ¨a et al., 1997; Kulmala et al., 1998b, 2001). By doing this, only the maximum in the particle size distribution is used, and, as is the case for the methods discussed above, the estimated growth rate is averaged over relatively long time scales, thereby masking variations in the growth rate.

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McMurry and Wilson (1982) determined the growth rate by solving the growth term in the cumulative form of the General Dynamic Equation (see Sect. 2.1). This same principle was used by Verheggen and Mozurkewich (2002), who first corrected the measured size distributions for coagulation and dilution before determining the growth rate by linear interpolation in a plot of consecutive cumulative size distributions.

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Since the current commercially available measurement techniques can only detect particles larger than 3 nm diameter, the nucleation rate is not directly measurable. In-stead, many studies have reported the appearance rate of particles above a certain threshold diameter, dictated by the minimum detectable size of the instrumentation used (Weber et al., 1995; O’Dowd et al., 1998). Often, the appearance rate is deduced

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from the increase in total particle concentration larger than 3 nm diameter (O’Dowd et al., 1998). Attempts have been made to relate this appearance rate to the actual nucleation rate by estimating the amount of particle losses since the time of nucleation due to coagulation and deposition (O’Dowd et al., 1999; Kulmala et al., 2001).

In this paper a novel method is described to accurately determine the empirical

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cle nucleation and growth rates. The particle growth rate is determined via a non-linear regression analysis of the General Dynamic Equation (GDE) (Friedlander, 2000) to fit the measured change of the aerosol size distribution in time. This way, the growth rate is determined using a range of size intervals rather than a total number or only the

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Printer-friendly Version Interactive Discussion imum of the distribution, as is implicitly done when fitting the evolution of the maximum

number density in a contour-plot of consecutive size distributions. Knowing the growth rate as a function of time enables an estimate of the time of formation of measured particles. By integrating the losses that have occurred between time of formation and time of measurement, the number density of nucleated particles can be determined.

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Knowing the concentration of nucleated particles and the time interval in which they formed gives the nucleation rate. This is different from other methods that are based on fitting the nucleation rate using an aerosol dynamics model (Lehtinen et al., 2004) or on correcting the appearance rate for coagulation (O’Dowd et al., 1999; Kulmala et al., 2001; Kerminen and Kulmala, 2002).

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The program, called PARGAN (particle growth and nucleation), is written using IGOR Pro software (Wavemeterics, Inc.) and will be described in detail below. Its application to measurements made in the Calspan environmental chamber will be discussed. This method can serve as a powerful tool to improve our understanding of nucleation by pro-viding data on nucleation in the atmosphere that do not depend on classical nucleation

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theory. These data could in turn be used to develop empirically based parameteriza-tions to the nucleation rate, for use in simulation modeling.

2 Theory

2.1 General Dynamic Equation

The General Dynamic Equation (GDE) describes the evolution of the aerosol size

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tribution in time. The rate of change in cumulative particle concentration, defined as the concentration of particles larger than a certain size, is used as the quantity being fit. This greatly simplifies the growth term, and it tends to dampen the effect of noise in

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Printer-friendly Version Interactive Discussion the data. The cumulative form of the GDE is given by

∂N_c(r_c) ∂t = r∞ Z rc  ∂n(r) ∂t dr = − r∞ Z rc (k_L(r)n(r))dr+ g(r_c)n(r_c)+ r∞ Z r_c   rc Z r₀ k_C(r₁, r₂)n(r₁)n(r₂) r r₂ 2! dr₁  d r − 2 r∞ Z r_c  n(r ) r∞ Z r₀ (k_C(r, r₁)n(r₁)) dr₁  d r (1)

where r₀and r_∞are the minimum and maximum detectable radii, respectively. The size distribution function is given by n(r)=dN(r)/dr, where N(r) is the number of particles

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of radius r per unit volume; N_c(r_c) denotes the cumulative number concentration of particles larger than r_c. Equation (1) is obtained by integrating the regular form of the GDE. For clarity of presentation, the time indexes have been omitted here, though it should be kept in mind that the size distribution function, n, and the particle radius growth rate, g, are a function of both radius, r, and time, t. The first term on the

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right hand side of Eq. (1) describes the effect of first order losses (e.g. deposition and dilution), where k_L(r) is the size dependent first order rate constant. The second term describes the effect of growth by condensation of low vapour pressure species. The third and fourth terms describe the effect of particles being produced by coagulation of two smaller particles (of radii r₁ and r₂, where (r₁)3+(r₂)3=r3) and particles being

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lost by coagulation with another particle (of radius r₁), respectively. The second order rate constant for coagulation of particles of radius r₁with those of radius r₂is given by

k_C(r₁, r₂). These processes will be discussed in more detail below.

Direct emissions also influence the ambient size distribution, which could be repre-sented by a zero order source term in the GDE, as it is usually independent of n(r).

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This has been omitted from Eq. (1). Nucleation is not explicitly included in Eq. (1), since with the currently available instrumentation, the minimum detectable size will be larger than the size of a critical cluster. However, its effect on n(r) is implicitly included

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Printer-friendly Version Interactive Discussion in the condensational growth term and the boundary condition at r₀, which describe

how the recently nucleated particles grow into the measured size range. 2.2 Condensation

The condensational growth rate, g(r,t), can be written as

g(r, t)= g₀(t)γ(r) (2)

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where g₀(t) is the radius growth rate in the gas kinetic limit, assuming a mass accom-modation coefficient of unity. Its time dependence is due to the change in concentration of the condensing species. γ(r) is an effective uptake coefficient (i.e. the inverse re-sistance; Molina et al., 1996) given by a rearrangement of the equation of Fuchs and Sutugin (1970): 10 1 γ(r) = 1 α + 3r 4λ_vap − 0.47r r+ λ_vap (3)

where α is the mass accommodation coefficient for the condensing species and λ_vapis the gas phase mean free path of the condensing vapour. λ_vapis only weakly dependent on the nature of the condensing species, and, for use in Eq. (3), is by definition

λ_vap≡ 3Dvap

ν_vap (4)

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where D_vapis the gas phase diffusion coefficient and ν_vapis the mean molecular speed of the condensing vapour.

The radius growth rate can be related to the concentration of the condensing species in terms of the uptake coefficient, γ(r), via

g(r, t)=dr d t =

γ(r)ν_vapM_W ([X ] − [X ]_sat)

4ρwN_A (5)

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Printer-friendly Version Interactive Discussion where M_W is the molecular weight of the condensing vapour, [X ] and [X ]_sat are its

actual and saturation concentration, respectively, ρ is the particle density, N_A is Avo-gadro’s number and w is the mass fraction in the particle of the condensing species.

As illustrated by Eq. (5), the net rate of condensation is proportional to the excess concentration of the condensing species above saturation. During periods of vigorous

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growth, the saturation concentration can be assumed negligible compared to the ac-tual concentration. The time dependence of the growth rate is due to the change in vapour concentration, while the size dependence is due to the size dependence of the uptake coefficient, γ(r). This size dependence disappears for small particles. When [X ][X ]_sat, then the Kelvin effect does not exert a significant size dependence on the

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net growth rate.

Condensation is a growth process for the particle and a loss process for the gas-phase condensing species. The pseudo-first order loss rate of the vapour due to con-densation (also called “concon-densation sink”) is given by

k_cond(t)= r∞ Z r0 γ(r)πr2ν_vapn(r, t)dr (6) 15

2.3 First order loss processes

Any first order loss process that reduces the particle concentration (e.g. deposition, dilution) can be included in the definition of k_L(r); thus, its definition depends on the processes that it describes. For smog chamber measurements, wall loss is the main first order loss process; this depends only on particle size, if the mixing is assumed

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constant in time. Two first order processes relevant for smog chamber studies are discussed here: wall loss by diffusional deposition and wall loss by gravitational settling. For small particles, wall loss by diffusion is most important, while for larger particles, loss by gravitational settling is larger.

A number of chamber studies (Crump and Seinfeld, 1981; McMurry and Rader,

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Printer-friendly Version Interactive Discussion 1985; Bienenstock, 2000; Hoppel et al., 2001) have reported a first order rate constant,

k_di_ff(r), for diffusional wall loss that is proportional to the square root of the Brownian diffusion coefficient, D_B(r). Thus, k_di_ff(r) is given by

k_di_ff(r)= C_di_ff q

D_B(r) (7)

where C_di_ff is a proportionality constant (in cm−1 s−1/2) and D_B(r) is the Brownian

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diffusion coefficient of a particle of radius r. The value of the proportionality constant,

C_di_ff, is dependent on the dimensions of and the amount of turbulence in the chamber, and its value can only be determined empirically.

The first order rate constant for wall loss by gravitational settling, k_grav(r), is given by

k_grav(r)= m_part(r)B(r)GS

VFs (8)

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where m_part(r) is the mass and B(r) is the mobility of a particle of radius r, G is the acceleration by gravity, S/V is the surface to volume ratio of the chamber, and F_s is the dimensionless ratio of projected horizontal surface area to total surface area in the chamber.

For atmospheric measurements, S/V is the inverse of the height of the planetary

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boundary layer, and F_sis unity. The effect of dilution can be included by using a suitable tracer (Verheggen and Mozurkewich, 2002).

2.4 Coagulation

The second order coagulation rate constants, k_C(r₁, r₂), are calculated according to Sceats (1989). Enhancement factors due to van der Waals forces are included, using

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a Hamaker constant of 6.4×10−20J, as determined from coagulation rates measured for H₂SO₄ (72% by mass) particles between 49 and 127 nm diameter by Chan and Mozurkewich (2001). Since these rate constants are defined so as to be consistent

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Printer-friendly Version Interactive Discussion with chemical kinetics conventions for reactions between identical particles, there is a

factor of two in the coagulation loss rather than a factor of 1/2 in the production term. When only scavenging by larger particles is considered, coagulation acts as a loss term and can be treated as pseudo first order. Then the pseudo-first order coagulation rate constant, k_C,I(r₁, t), is given by

5 k_C,I(r₁, t)= r∞ Z r0 k_C(r₁, r₂)n(r₂, t)dr₂ (9)

3 Obtaining growth rates

Normally, the GDE would be used to calculate the evolution of particle size distributions using input parameters such as growth rate and coagulation rate constants. Here we used the measured change in the cumulative size distribution over a finite time interval,

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∆Nc(rc)/∆t, as an input variable, and use non-linear regression analysis to determine

one or more parameters, such as the growth rate. Thus, this is a form of inverse modeling.

Specifically, in this work we focus on determining the gas kinetic growth rate, g₀(t), by fitting ∆N_c(r_c)/∆t as a function of particle size. By applying this procedure to a

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series of consecutive time intervals, the growth rate as a function of time is obtained. Effects of other processes, such as deposition or wall loss, dilution, and coagulation can also be investigated by modifying their assumed values or by fitting some of them as part of the regression analysis.

To apply the GDE to discrete size distribution data, the differentials of Eq. (1) are

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approximated by finite differences, both in terms of the time interval between two mea-surements, and the size difference between two neighbouring size bins. The integrals are evaluated as straight summations. In the following, r and r_cstand for the radius at the centre of a size bin, and∆r is the size bin width. All contributions to ∆N_c(r_c)/∆t

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Printer-friendly Version Interactive Discussion and to n(r) are evaluated at the bin centre, and are assumed to be constant over the

size bin. Each SMPS scan takes a finite amount of time. The measurement time for a scan is taken to be the time at which the SMPS detected the maximum in the number density.

The two coagulation terms have one fitting parameter; this multiplies the production

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and loss terms by a single value, and thus always conserves volume, but it does not allow for any variation in the size dependencies of the coagulation rate constants. Both coagulation loss and production are included in the regression analysis. These pro-cesses are often either ignored or treated only as a loss process for particle number. However, their effect is often significant, even when the size distributions are narrow

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and mono-modal. Coagulation with particles smaller than the minimum detectable ra-dius is not included in the determination of the growth rate, because their concentration is not known. This may lead to an overestimation of the condensational growth rate if coagulation with those undetected particles contributes to the overall growth of the measured particles. For the application to smog chamber data, the first order losses

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include wall losses by diffusion and by gravitational settling; each is proportional to a single parameter (C_di_ffand density). For the condensation term, g₀(t) (see Eq. 2) is the default fitting parameter. Other parameters, such as α or D_vap, could also be fit, but this is only useful if the particle size range is wide enough for there to be a significant diffusion limitation to the uptake coefficient. We can assume that g(r_∞)n(r_∞)=0,

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vided that the concentration at r_∞ is very small compared to the concentration at the distribution maximum.

In Eq. (1), the change in particle number over an infinitesimal time interval,

∂N_c(r_c)/∂t, is expressed in terms of n(r, t), the size distribution at time t. However, in the finite difference approximation, the left hand side of Eq. (1) becomes the rate of

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change in particle number from time t₁ to time t₂. Then n(r, t) is no longer precisely defined; it has to be approximated as an average of n(r, t₁) and n(r, t₂). The most common way of averaging amounts to taking the average value of n(r, t₁) and n(r, t₂)

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Printer-friendly Version Interactive Discussion at constant radius, r (e.g. McMurry and Wilson, 1982):

n(r, t) ≈ n(r, t1)+ n(r, t2)

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Since the number density is a function of both time and radius, Eq. (10) represents an approximation to, and not the definition of, n(r, t). For example, consider a narrow distribution that grows so rapidly that the change in size between two successive

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surements is greater than the width of the distribution. Then Eq. (10) gives a bimodal distribution with peaks near the initial and final modal sizes. But the correct average, which satisfies the GDE, is a mono-modal distribution with a peak between the initial and final modal sizes and spread over the full range of initial and final sizes.

From sensitivity studies with synthetic data, we conclude that when the radius growth

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over a single time interval is less than (3×σ) nm (equivalent to a growth rate of 54 nm h−1, for a relative geometric standard deviation of σ=1.5 and a 5 minute scan time), the error in the growth rate is less than 5%. Therefore the approximate average as defined by Eq. (10) is used in the regression analysis.

3.1 Finite difference approximation of the coagulation term

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The contribution of coagulation loss to∆N_c/∆t is determined by numerically

evaluat-ing the integrals in Eq. (1). The coagulation production term is less straightforward, because of the Jacobian factor, (r/r₂)2 (Williams and Loyalka, 1991). This factor ac-counts for the fact that the radius of the produced particle is not the sum of the radii of the two coagulating particles. It is difficult to evaluate numerically because the

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ness of the size bins makes it impossible for all the initial and final radii to correspond to bin centres. This difficulty can be avoided by recognizing the physical basis of the Jacobian factor: It describes the way that the particles produced by coagulation are spread out over the distribution. In a numerical calculation it is more natural to simply consider all possible pairs of bins and to distribute, for each pair, the coagulation

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ucts between bins in a manner that conserves both number and mass. This distribution 1690

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Printer-friendly Version Interactive Discussion between bins is equivalent, for a discrete distribution, to applying the Jacobian factor.

In our procedure, we choose the sizes r₁and r₂of the two coagulating particles to be at bin centres, named r_i and r_j, respectively, to distinguish them from usage in the con-tinuous version of the GDE (Eq. 1). The produced particles of radius r=((r_i)3+(r_j)3)1/3 are divided over two neighbouring size bins using a procedure that is similar to those

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used in forward modeling exercises (Toon et al., 1988; Jacobson et al., 1994). A frac-tion, f_c, of these particles is assigned to the size bin with centre radius r_high>r, and a

fraction (1-f_c) is assigned to the bin with centre radius r_low<r. The centre radius of the

smallest size bin to be included in the cumulative number concentration is denoted by

r_c. Then, in the discrete form of Eq. (1), the rate of change of the cumulative distribution

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due to the coagulation production term becomes ∆Nc(rc) ∆t = i=bin∞ X i=bin0   j=bin∞ X j=bin0 f_c(r_c, i , j )k_C(r_i, r_j)n(r_i)n(r_j)∆r_i∆r_j   (11) where f_c= 0 if r_c < r_low (12a) f_c= r 3 − r_low3

r_high3 − r_low3 if rlow< rc< rhigh (12b)

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f_c= 1 if r_c > r_high (12c)

Particles are not allowed to be formed outside the measured size range; this requires that the size distribution extends well beyond the size range for which concentrations are significant. This contributes to internal consistency and offers a useful quality con-trol check, in that the total coagulation loss should equal twice the total coagulation

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Printer-friendly Version Interactive Discussion 3.2 Weighted fitting of the GDE to the measured data

To decrease the effects of noise in the data, more accurate measurements carry more weight in the regression analysis than less accurate measurements. Via an iterative procedure, the sum of the weighted squared differences (χ2) between fitted and mea-sured value of∆N_c/∆t is minimized for all measured size bins simultaneously, so the

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resulting value of the growth rate (and/or other fitting parameters) is the optimum value taking into account all size bins (or a specified sub-range of bins). χ2is defined as:

χ2= r∞ X r0 _y fit(rc) − ymeas(rc) σ_meas(r_c) 2 (13)

where y_fit(r_c) is the fitted value, y_meas(r_c) is the measured value, and σ_meas(r_c) is the standard deviation of∆N_c(r_c)/∆t. PARGAN uses the non-linear least-squares

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dure in Igor Pro to search for the parameter values that minimize χ2. Igor Pro uses an implementation of the Levenberg-Marquardt algorithm, as described by Press et al. (2002). If the values of the standard deviations are good estimates of the actual er-rors, then χ2is of the same order of magnitude as the number of degrees of freedom; larger values of χ2indicate that additional sources of error were present.

15 We estimate σ(r_c) by: σ(r_c)= "_i_=bin ∞ P i=bin_rc n (∆ri) 2

2n2_min(ri)+ (none(ri)n(ri, t1))+ (none(ri)n(ri, t2))+ (Qn(ri, t1)) 2

+ (Qn(ri, t2)) 2o

#1₂

t₂− t₁ (14)

where n_min(r_i) is the concentration, d N(r_i)/dr_i, at size bin i corresponding to the mini-mum incremental number of particle counts, n_one(r_i) is the concentration corresponding

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to one measured count in size bin i , and Q is an empirical, size independent, dimen-sionless constant, discussed in the following. Here t₁ and t₂ are the times at the

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Printer-friendly Version Interactive Discussion beginning and end of the time interval∆t. The summation in Eq. (14) is required since

this is the standard deviation of a cumulative concentration for all size bins larger than the one centred around r_c.

The first term in Eq. (14) describes the variance due to measuring only the minimum increment of counts. Depending on how the counts are measured, this increment may

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be greater than unity; for the Calspan measurements, this minimum increment was 14 counts. This term only contributes significantly to σ(r_c) when the number of counts in the size bin is small. It accounts for the fact that a measurement of zero counts is not infinitely accurate and prevents the weight from becoming infinity when zero counts are recorded. The factor of 2 accounts for the fact that this term, as the others, is included

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for the measurements at both ends of the time interval.

The next two terms describe the usual variance due to counting statistics, for each measurement time, σ(r_i)=n(r_i)/√counts. The last two terms in Eq. (14) describe the variance due to uncertainty in flow rates or other sources of error that are directly proportional to the concentration. The value of the constant, Q, was determined

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pirically. At high concentrations, variations in concentration are typically 1% to 2%; this is much larger than expected from counting statistics. For the Calspan experiments, Q was chosen to be 0.01, based on obtaining the correct order of magnitude for the value of χ2. The error bars that we report for fitting parameters are based on the scatter around the fit.

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4 Application to smog chamber measurements: wall loss, coagulation, and

growth rate

The measurements were conducted in Calspan’s 590 m3environmental chamber dur-ing October and November 1998; Details of the chamber and its instrumentation, along with results from selected case studies on the ozonolysis of α-pinene are given by

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pel et al. (2001). To illustrate the use of PARGAN, we apply it to an experiment in which SO₂oxidation resulted in particle nucleation and growth.

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Printer-friendly Version Interactive Discussion 4.1 Chamber characteristics and experimental methods

The Calspan chamber has a total volume of 590 m3with a diameter and height of both 9.1 m. This provides a surface to volume ratio S/V of 0.67 m−1and a relative projected horizontal surface area F_s of 0.167. It has a large mixing fan and the interior is teflon coated. A filtration system lowers measured gas phase and aerosol concentrations to

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below detectable levels by overnight filtration. Prior to each experiment, the chamber was filtered overnight, then sealed while background particle and gas phase concen-trations were monitored for one hour. Air removed from the chamber for sampling was replaced through activated charcoal and absolute particle filters.

The aerosol size distribution from 4.4 to 404 nm radius was measured using a NRL

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DMA and MetOne 1100 CNC in scanning mode. The filtered and re-circulated sheath air of the DMA was dried and the aerosol sample was removed from the chamber through a diffusion dryer. A complete scan was measured every 288 s. The measured size distributions were corrected for particle losses in the sample lines and for reduced CPC counting efficiency at small particle sizes prior to data analysis.

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4.2 Determination of the wall loss and coagulation rates

The growth and nucleation rates determined by PARGAN depend on the values used for the wall loss and coagulation rate constants. Those constants can be determined by applying PARGAN to experimental data obtained under conditions under which no particle growth was occurring.

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Although coagulation rate constants may be calculated theoretically, there is con-siderable uncertainty in the results. To account for this, we introduce a dimensionless parameter, the coagulation multiplier, C_coag, by which the theoretical coagulation rate constants are multiplied in order to agree with the measurements. PARGAN enables the determination of both C_coag and the proportionality factor for the wall loss (C_di_ff in

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Eq. 7) by means of regression analysis. The rate of change of total particle volume and number can also be used to determine these values; the two methods give consistent

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Printer-friendly Version Interactive Discussion results.

Both C_coag and C_di_ff are determined from an experiment conducted on 16 Novem-ber 1998. The oxidation of gas phase SO₂ induced particle nucleation and growth in the absence of pre-existing aerosol. The size distribution was allowed to evolve in the dark, during which time coagulation was the dominant process occurring. The air in

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the chamber was then circulated through a filter to produce a substantial reduction in number concentration to around 1.3×104cm−3; after this, both wall loss and coagula-tion were expected to be important. This last segment of the experiment was used in the analysis described in the following.

First, the rate of change in particle volume and number are used to provide an

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timate of the wall loss and coagulation rates. In the absence of condensation, the decrease of particle volume is caused solely by wall losses, since coagulation does not alter the total volume. To filter out the effect of random counts in the larger size bins, only contributions from particles below 50 nm are included in determining the measured particle volume. This includes the entire distribution, which had number and

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volume maxima at 14 and 17 nm radius, respectively. Since these are ultrafine parti-cles, gravitational settling can safely be ignored. Figure 1 shows the measured particle volume and number as a function of time. Fitting Eq. (7), integrated over the distribu-tion, to the observed decrease in particle volume over the time interval from 14:30 to 15:15 yields C_di_ff=3.6×10−3cm−1s−1/2.

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The decrease in particle number is caused by both wall loss and coagulation. By assuming that there is no condensation growth, that the wall loss is given by Eq. (7) with C_di_ff=3.6×10−3cm−1 s−1/2, and that the coagulation rate constants are as calcu-lated except for a common unknown multiplier (C_coag), the GDE can be numerically integrated with respect to size. This yields an equation that depends on C_coagand

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vides the change in number concentration over any given time interval. Fitting this to the observed decrease in particle concentration from 14:30 to 15:15 yields C_coag=1.6.

Using these estimates of C_di_ffand C_coagas inputs in PARGAN gives a best fit average growth rate of −0.02±0.32 nm h−1, as expected since no condensation should have

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Printer-friendly Version Interactive Discussion occurred during this segment of the experiment. Regression analysis using PARGAN

can also be used to determine these parameters; however, it proved to be impossible to simultaneously fit both of these parameters and the growth rate. When the conden-sational growth rate was set to zero, the best fit yielded C_di_ff=3.7×10−3cm−1s−1/2and

C_coag=1.5. These values are in excellent agreement with the results from inspecting

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the rate of change in particle volume and number, discussed above.

To examine the effect of ignoring coagulation, we set C_coag=0. This yielded

C_di_ff=8.5×10−3cm−1 s−1/2, significantly larger than when coagulation was included. The same value was found when the decrease in particle number was fit by an ex-ponential decay while ignoring coagulation. This shows that the wall loss rate can be

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significantly overestimated when coagulation is neglected, as is often done in chamber experiments. Note that the number concentration decay plotted in Fig. 1 appears to be first order in spite of the significant effect of coagulation.

In the subsequent data analysis the wall loss by diffusion is given by

k_di_ff=(3.6×10−3cm−1s−1/2)×D_B1/2. The theoretical coagulation rate constants are

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tiplied by a factor of 1.5.

The use of a single coagulation multiplier is somewhat unrealistic in that deviations from calculated coagulation rate constants might be expected to vary with particle size. Sensitivity studies indicate that the fit is improved at small particle sizes by using even larger values of C_coag. However, the data are insufficient to draw any firm conclusion

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about its size dependence. We have no explanation for the surprisingly large coagula-tion rate constants found in this analysis.

4.3 Determination of the growth rate

In the experiment chosen for this example, 20 ppb of CH₂O and 1.2 ppb of NO were injected to generate OH radicals. At 10:55, 75 ppb of SO₂ was injected. Half of the

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chamber lights were turned on at 11:09; this started the photochemical production of OH and subsequent oxidation of SO₂ to H₂SO₄. At 11:59, more SO₂ was injected,

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Printer-friendly Version Interactive Discussion raising the concentration from 36 to 89 ppb, and all chamber lights were turned on.

This was followed by another injection of 20 ppb of SO₂and 2 ppb of NO at 12:50. The relative humidity during the experiment ranged between 75 and 82%. Binary nucle-ation of H₂SO₄ with water vapour produced new particles that subsequently grew by condensation and coagulation. Figure 2 shows the evolution of the particle number

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size distribution. This nucleation and growth experiment ended when the lights were turned off at 13:37 to determine wall losses.

The mass accommodation coefficient, α, and the diffusion coefficient of the condens-ing species, D_vap, both of which are included in the apparent uptake coefficient, γ(r), were held constant at 1 and 0.1 cm2 s−1, respectively (Jefferson et al., 1997; P¨oschl

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et al., 1998). The saturation concentration of the condensing species was assumed to be zero. For the experiments investigated, the magnitude of α, D_vap, and saturation vapour pressure can not be realistically verified by regression analysis since the fits were insensitive to these parameters. A size dependence of the growth rate due to Kelvin effect was not discernible.

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Results for the growth rate, obtained from regression analysis using PARGAN, are shown in Fig. 3. No growth rates could be determined prior to 11:45 since the freshly nucleated particles had not yet grown into the DMA size range. The growth rate can be related to the concentration of H₂SO₄ vapour by using Eq. (5). The values for ρ and w were determined by interpolating in the tables of Gmitro and Vermeulen (1964).

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Assuming fast equilibration at the low relative humidity (7.5%) inside the DMA, w was found to be 67% and ρ was 1.57 g cm−3. Under these conditions the growth rate in the kinetic limit (assuming α=1) is given by:

g₀(t)= 0.86 × [H₂SO₄(t)] (15)

with [H₂SO₄(t)] in pptv and g₀(t) in nm h−1. Figure 3 gives both the growth rate and

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the equivalent H₂SO₄concentration according to Eq. (15).

As can be seen in Fig. 3, the growth rate increased immediately after the SO₂ injec-tions. The maxima in the growth rate lag behind the SO₂injections by 5 to 10 min; this

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Printer-friendly Version Interactive Discussion is due to the time required for the H₂SO₄concentration to reach steady state. The time

to reach steady state can be approximated by the lifetime of H₂SO₄, defined as the inverse of the pseudo-first order loss rate for condensation and wall loss. At the start of the experiment, the lifetime was around 15 minutes; it decreased to approximately 4 min at 12:50 due to the increase in particle surface area. These lifetimes agree

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sonably well with the observed time delay between SO₂injection and the maximum in the growth rate.

The maxima in the growth rate that are clearly visible in Fig. 3 are not discernible from the contour plot of consecutive size distributions (Fig. 2). We found similar results for other case studies. This suggests that PARGAN allows the determination of the

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growth rate with much better time resolution than the common method of fitting a curve through the banana-shaped contour plot of consecutive size distributions.

Although the growth rate found by regression analysis depends on to the values cho-sen for the coagulation and wall loss rate constants, the cho-sensitivity is not great for the case investigated here. The growth rate changes by less than 10% if the coagulation

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multiplier is changed from 1.5 to 1.0 or if the wall loss rate is doubled.

Investigating other case studies from the Calspan measurements has shown that the concentration of the condensing species, as deduced from the growth rate via Eq. (5), agrees well with calculations using a simple chemical box model. This shows that an accurately known growth rate, as determined using PARGAN, can provide valuable

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information about the gas phase and heterogeneous chemistry of the system under investigation (Verheggen, 2004).

5 Obtaining nucleation rates

As a group of newly nucleated particles grow in size, their concentration changes as a result of coagulation and wall loss. We use the term “cohort” to refer to such a group

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of particles that are formed at approximately the same time. The determination of the nucleation rate is based on following this process backwards in time to sizes that are

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Printer-friendly Version Interactive Discussion smaller than the minimum measured size. Starting from the measurement time, the

change in cohort radius during each previous time step is determined from the growth rate. The evolution of cohort radius and number density backwards in time is evaluated for each measured size bin at each measurement time. The time when the backwards calculated radius equals the assumed radius, r_N, of the critical cluster (0.5 nm radius,

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following Weber et al., 1997, and Kulmala et al., 2000) is taken to be the time of particle formation, t_N.

The number density of the cohort is determined for each previous time step by inte-grating the losses that occurred in the time interval. The nucleation rate, J , is defined as the rate at which particles grow past the radius of the critical cluster. The number

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of newly nucleated particles, n(r_N, t_N), is related to the nucleation rate and the growth rate via (Weber et al., 1995; Verheggen and Mozurkewich, 2002)

J= n(r_N, t_N)g(r_N, t_N) (16)

In order to evaluate the time of formation and the nucleation rate, the growth rate has to be known for all time intervals between measurement time and formation time. The

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growth rate can only be determined from measurements when at least part of the size distribution is measured. For atmospheric measurements, these growth rates can only be used to determine the formation time and nucleation rate when the measured size distributions reflect the same air mass as that where nucleation actually took place. If those conditions are not met, but the concentration of condensing species (e.g. H₂SO₄)

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is known or can be estimated from time of formation onwards, theoretical growth rates can be calculated, and used to determine the time of formation and nucleation rate (Verheggen and Mozurkewich, 2002).

5.1 Evaluation of number density backwards in time

The loss terms of the GDE are applied to a cohort of particles, as they grow in size.

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The concentration of particles at the critical cluster size is determined by numerically integrating the pseudo-first order losses that occurred between the time of nucleation

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Printer-friendly Version Interactive Discussion (t_N with radius r_N) and the time of measurement (t_M with radius r_M). To this end, the

measured particle number density is multiplied by a correction factor for wall losses by diffusion, DF , and a correction factor for coagulation scavenging, CF :

n(r_N, t_N)= n(r_M, t_M) × DF × CF (17) where 5 DF = exp    tM Z tN k_di_ff(r, t)d t    =exp   rM Z rN k_di_ff(r, t) g(r, t) dr   (18) and CF = exp    tM Z tN k_C,I(r, t)d t    =exp   rM Z r_N k_C,I(r, t) g(r, t) dr   (19)

where k_di_ff(r, t) is the first order rate constant for wall losses by diffusion (Eq. 7) and

k_C,I(r, t) is the pseudo first order rate constant for coagulation scavenging (Eq. 9). The

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time dependence of k_di_ff(r, t) is due to the cohort radius changing in time. A different procedure is followed to obtain an additional correction factor for within-mode coagula-tion; this is described in Sect. 5.4.

These correction factors are equivalent to evaluating the GDE, without the growth term, as a total derivative with respect to time; the effect of growth is implicitly included

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via the change in cohort radius. The detailed evaluation of the correction factors will be described in subsequent sections. The correction factors can be thought of as the fractional decrease in the number density from time of nucleation to time of measure-ment. Note that Eqs. (17) through (19) are applied to a cohort of particles as they grow in size; thus both r and t are changing. The final correction factors, DF and CF , are

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the product of the individual correction factors, DF_i and CF_i, which are evaluated over the cohort radii between consecutive measurement times. This has the advantages

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Printer-friendly Version Interactive Discussion that the growth rate can be considered constant over each time interval so that it can

be taken out of the integral. Also, it enables the evaluation of correction factors for any time and/or size, an example of which is described next.

The extrapolated number concentration at a time, t, may be obtained by replacing

t_N with t_h in Eqs. (17) through (19). This allows the characterization of the size

dis-5

tributions as they are predicted to have existed at previous time steps. The resulting “reconstructed” size distributions can be compared to the measured size distribution for the same time interval. They should all closely resemble each other if the extrapolations are accurate and internally consistent. An example of such a collection of reconstructed size distributions is given in Fig. 4. The measured distribution is matched reasonably

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well, especially in light of the extrapolations on which this calculation is based. The ap-proximate treatment of within-mode coagulation is not included in these reconstructed size distributions. Figure 4 also illustrates the determination of the number of nucle-ated particles: The concentration at a particle radius of 0.5 nm (the assumed size of the critical cluster) provides the number density at the time of nucleation used in Eq. (16).

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5.2 Correction for wall losses

In the kinetic limit, the diffusion coefficient varies as 1/r2; thus, Eq. (7) becomes

k_di_ff(r)=C/r. Using the result of Sect. 4.2 gives C=4.1×10−4nm s−1. For one time step, the correction factor for deposition, DF_i, can be solved by direct integration

DF_i = exp    r(t₂) Z r(t1) k_di_ff(r) g dr    =exp    C g r(t₂) Z r(t1) 1 rdr    = _r(t 2) r(t₁) Cg (20) 20

where t₁<t₂. Since the growth rate, g(r,t), is assumed constant for the time interval from t₁to t₂, and independent of size for the small change in cohort radius from r(t₁) to

r(t₂), it can be taken out of the integral and is written as g instead. The total correction factor for wall losses is obtained by multiplying the individual correction factors for each

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Printer-friendly Version Interactive Discussion interval between r_M and r_N. The first equality in Eq. (20) is generally applicable to any

first order process, whereas the second and third equality are specific to wall losses of small particles in smog chamber studies.

5.3 Correction for coagulation scavenging

As particles grow from the critical cluster size to the measured size they undergo

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ulation. We divide the coagulation events that might occur into two subsets. First, we have the coagulation of the growing cohort of particles with particles that are larger than both the cohort size and the minimum measured size. We call this “coagulation scav-enging” since it is a first order loss process for the growing particles. Since we know the size distribution of the larger particles, we can explicitly calculate a pseudo-first

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der rate constant for the loss of particles from the growing cohort. The second subset consists of coagulation of cohort particles with particles of comparable or smaller sizes. This “within-mode coagulation” is more difficult to treat. We describe our treatment of coagulation scavenging in this section and treat within-mode coagulation in the next section.

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The correction factor for coagulation scavenging, CF_i, over the time interval from t₁ (with cohort radius r₁(t₁)) to t₂(with cohort radius r₁(t₁)) is written as

CF_i = exp    1 g r1(t2) Z r1(t1)    r∞ Z r₀0 (k_C(r₁, r₂)n(r₂)) dr₂   dr1    (21)

where k_C(r₁, r₂) is the second order rate constant for coagulation and where

r₀0 = r₁(t₂) if r₂> r₀ (22a)

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r₀0 = r₀if r₂< r₀ (22b) Equation (21) is directly analogous to Eq. (19) where the inner integral is the pseudo-first order rate constant for scavenging of cohort particles of radius r₁by all larger par-ticles of radius r₂, provided that they are within the measured size range. Performing

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Printer-friendly Version Interactive Discussion numerous evaluations of the double integral in Eq. (21) is cumbersome. The following

procedure can be used to obtain an approximate expression that involves only a sin-gle integral. Since the inner integral is independent of the cohort radius the order of integration can be exchanged to obtain

CF_i = exp    1 g r∞ Z r₀0   n(r2) r1(t2) Z r1(t1) (k_C(r₁, r₂)) dr₁   dr2    (23) 5

In order to evaluate this integral analytically, the second order rate constant is fit to a power law of the form k_C(r₁, r₂)=h(r₂)+c(r₂)×r₁p(r2). There is a separate set of param-eters (h, c, and p) for each discrete value of the larger radius, r₂ (corresponding to the bin centres). This power law can be analytically integrated, after which the outer integral is numerically integrated. This yields

10 CF_i = exp    1 g j=bin_∞ X j_=bin0 0 h_j(r₁(t₂) − r₁(t₁))+ cj p_j + 1 r₁(t₂)(pj+1) − r₁(t₁)(pj+1) n(r_j)∆r_j    (24)

The distribution function n(r_j) is evaluated as the average between t₁and t₂. The total correction factor for losses by coagulation scavenging is obtained by multiplying the individual correction factors for each interval between r_M and r_N.

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5.4 Correction for within-mode coagulation

The correction described in the previous section only accounts for coagulation with particles larger than the growing cohort. Coagulation between particles of comparable sizes both reduces the number and increases the size. We treat this “within mode” coagulation approximately using a first order perturbation method. Two different

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Printer-friendly Version Interactive Discussion assumes that growth and nucleation are in steady state, and thus constant, between

the time of nucleation and the time of measurement. Consequently the size distribution function in the nucleation mode has a maximum at the critical cluster size, decreases with increasing size, and does not change with time. Nucleation may approach steady state when it is prolonged in time due to a reasonably constant super saturation of

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vapour.

The second limiting case assumes that there is a sharp maximum in the nucleation rate (a nucleation pulse) followed by growth of the nucleated particles. Consequently the size distribution function in the nucleation mode has a maximum at a size larger than that of the critical cluster. The pulse model assumes that the distribution consists

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of a cohort of identically sized particles, with a total number density of N. A nucleation pulse may occur due to a transient increase in vapour concentration to a value above some threshold. This is especially likely in a smog chamber, where the temperature and relative humidity are relatively constant, and more precursor gas is periodically injected.

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The mathematical derivation of the limiting case of steady state nucleation is given in Appendix A; the correction factor is given by

n(r_N) n(r_M) = b1(rN)  1+ n(r_M) rM Z rN q(r) b₁(r)dr   (25) where b₁(r)= exp   1 g rM Z r k_di_ff(r0)+ k_C,I(r0) dr0   (26) 20 and q(r)=2b1(r) g rM Z r_N k_C(r, r₁)b₁(r₁)dr₁−2 g rhalf Z r_N k_C(r₁, r₂)b₁(r₁)b₁(r₂) r r₂ 2 dr₁ (27) 1704

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Printer-friendly Version Interactive Discussion where (r₁)3+(r₂)3=r3 and g is the growth rate, which is assumed to be constant and

independent of size for the nucleation mode particles. The second integral is sym-metrical with respect to exchange of r₁ and r₂; therefore, instead of evaluating it up to r, it is evaluated as twice the integral up to r_half=(r3/2)1/3. This avoids numerical instability by preventing the Jacobian factor from approaching infinity. Note that b₁(r_N)

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is the correction factor for pseudo-first order losses, DF ×CF (Eqs. 22 and 23). The factor within brackets in Eq. (25) is the correction factor for within-mode coagulation, assuming steady state nucleation.

All integrals, except the one in Eq. (26), are determined numerically. Naturally, there is a trade-off between the accuracy of the solution and computing time. A size spacing

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for numerical integration of 0.5 nm gives satisfactory results; reducing the step size to 0.01 nm changes the results by less than 10%.

To save computing time in the repeated calculation of b₁(r), an approximate, analytic integration is used. The wall loss rate constant is given by Eq. (20) and the pseudo-first order rate constant for coagulation is fit to the power law k_C,I(r, t)=h(t)+c(t)×rp(t),

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where the fitting parameters are a function of measurement time, t, and r is the cohort radius. Substituting these functions into Eq. (26) yields

b₁(r)= _r M r C_g × exp 1 g h ((r_M − r)+ c p+ 1 r_M(p+1)− r(p+1) (28)

The growth rate and the fitting parameters are averaged over the integration interval, i.e. from the time when the cohort radius was r to the time of measurement, when the

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cohort radius was r_M. A logarithmic average was used, since this was found to give good agreement between b₁(r) and the first order correction factors, CF ×DF .

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Printer-friendly Version Interactive Discussion Appendix B. The correction factor is given by

N(r_N) N(r_M) = b1(rN)  1+ N(r_M) rN Z rM p(r)dr   (29)

where b₁(r) is given by Eq. (26) and

p(r)= kC(r, r)b1(r) g2 1 3rkL,I(r) − g (30)

where k_L,I(r) includes (besides wall losses) coagulation scavenging by particles with

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radii greater than the cohort radius, r, treated as a pseudo-first order process; k_C(r, r) refers to coagulation with particles of the same size range only. The quantity within brackets in Eq. (30) is the correction factor for within-mode coagulation, assuming a pulse of nucleation. Here N(r_M) is the total number of particles in the narrow nucleation mode. As in the steady state case, b₁(r) is calculated according to Eq. (28) and all

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other integrals are evaluated numerically.

6 Determination of the nucleation rate in the smog chamber

We now illustrate this procedure by estimating the nucleation rates corresponding to the data in Figs. 2 and 3. Nucleation rates can only be determined when the growth rate is known, that is, for the period after 11:40. Two distinct particle modes can be

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seen in Fig. 2. The first mode seems to consist of particles formed following both the first and second SO₂ injections; the CPC readings show that nucleation bursts followed both injections. However, the particles formed in the first burst did not reach sizes detectable by the DMA until shortly before the second SO₂injection.

Calculated nucleation rates (only including the measured size bins which form the

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first mode) are shown in Figs. 5a and b. The former excludes the effects of within mode coagulation, whereas the latter includes these effects, assuming a pulse of nucleation.